This covers all transformations of functions by integrals, including but not limited to the Radon, X-Ray, Hilbert, Mellin transforms. Use (wavelet-transform), (laplace-transform) for those respective transforms, and use (fourier-analysis) for questions about the Fourier and Fourier-sine/cosine ...

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47 views

The resolvent kernel

Good morning. How can we prove that $$ \Gamma (s,t, \Lambda ) - \Gamma (s,t, M ) = \Lambda - M \times \int { \Gamma(s,t, \Lambda ) \times \Gamma (s,t,M) \, dx }$$ I tried to use the formula of ...
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0answers
49 views

Radial Green's function

I would like to solve an equation of the form $$ \bigg(\frac{d}{dr^2} + m^2 \bigg)f(r) = g(r), $$ for $f(r)$. Normally I would just find the Green's function $G(r,r')$, which is defined by $$ ...
2
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1answer
128 views

Fourier transforms and Dirac delta function

What is the Dirac delta function $\delta(t_1-t_2)$ in Fourier (frequency) space?
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1answer
26 views

How do you differentiate a Laplace transform?

Consider the Laplace transform of $\color{green}{t\cfrac{\mathrm{d^2}f}{\mathrm{d}t^2}}$: ...
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20 views

Fourier transform involving completion of square

Show that the Fourier transform of $$f(x) = k e^{\frac{-x^{2}}{2\sigma^{2}}}$$ for some $k \in \mathbb{R}$ is $$F( \lambda) = \frac{k}{\sigma}e^{\frac{-\sigma^{2}}{2\sigma^{2}}}$$ By ...
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1answer
67 views

Laplace Transform

The question I had was Find the Laplace transform of $$f(t)=10e^{-200t}u(t).$$ Would it be correct to take out the 10 because it is a constant, find the Laplace transform of $e^{-200t}$ and then ...
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0answers
16 views

Abel transform of $R_z(\Delta)$

The spectrum of the Laplace operator $\Delta$, as self-adjoint unbounded operator on $L^2$ is equal $]-\infty ,0]$. The resolvent $R_z$ is defined for $z \in \mathbb C \setminus ]-\infty ,0] $. Using ...
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1answer
75 views

Mellin transform of Gumbel distribution

The probability density function (PDF) of Gumbel distribution is given as: $$f\left(x\right)=\frac{\exp \left(-\left(\exp \left(-\frac{x-\mu}{\beta }\right)+\frac{x-\mu}{\beta }\right)\right)}{\beta ...
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12 views

Limitations on Linear Canonical Transforms

Preface: LCTs are used in physics and in signal processing, but this question is about the mathematics behind them moreso than any particular application. As such, I think it belongs here, but if I ...
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18 views

For a good understanding of Gauss' theorem

Why does any term that can be written as a divergence vanish when integrated over the volume V? And what is the physical meaning of this property? For example, $$\int_V\frac{\partial}{\partial ...
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0answers
27 views

A specific partial differential equation using Fourier Transform

I have the following PDE problem which I think sounds like a job for the Fourier transform: $ u_t + 2u_x = u_{xx} \space \space \space -\infty < x < \infty \space \space \space t>0 $ ...
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0answers
26 views

Hilbert transform of $\cos(\phi(t))$.

I am attempting to derive the Hilbert transform of $\cos{\phi(t)}$. I understand that the transform is given by \begin{align*} H[\cos(\phi(t)] = \frac{1}{\pi} \ p.v. \ \int_{-\infty}^{\infty} ...
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0answers
31 views

Nonstandard analysis and integral transforms

Can integral transforms be evaluated without limits(i.e Laplace transform) such as in non standard analysis? Can the improper integral be bounded by a hyperreal number? I am not very familliar with ...
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0answers
58 views

Finding the inverse of an Integral Transform

I'm working with the following integral transform - $\tilde{f}(y) = \int_{-\infty}^\infty dx\,\frac{f(x)}{x + i y}$ What would be the inverse of a such an integral transformation?
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3answers
94 views

Integrate $\int\sqrt{x+\sqrt{x^{2}+2}}$ $dx$ .

Q) $\int\sqrt{x+\sqrt{x^{2}+2}}$ $dx$ . Tried rationalising the numerator twice to get Numerator =-2 but not able to simplify denominator The question reduces to (as per my rationalising)
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1answer
38 views

Integrate $\int\dfrac{x \cos x+1}{\sqrt{2x^{3}e^{\sin x}+x^{2}}}dx$ [closed]

$$\int\dfrac{x \cos x+1}{\sqrt{2x^{3}e^{\sin x}+x^{2}}}dx$$
6
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1answer
76 views

Integrate $\int\frac{x^{6}-2x^{3}}{\left(x^{3}+1\right)^{3}}dx$ .

$$\int\frac{x^{6}-2x^{3}}{\left(x^{3}+1\right)^{3}}dx$$ I tried adding and subtracting 1 to bring a square expression with numerator as $(x^3-1)^2 -1$ but always going to partial fraction which ...
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1answer
42 views

Integrate $\int\frac{\left(1+x\right)\sin x}{\left(x^{2}+2x\right)\cos^{2}x-\left(1+x\right)\sin2x}dx$

(Q) $\displaystyle \int\frac{\left(1+x\right)\sin x}{\left(x^{2}+2x\right)\cos^{2}x-\left(1+x\right)\sin2x}dx$ Tried a lot to expand denominator and reduce it to bring its derivative on top , but all ...
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3answers
52 views

Integrate $\int\frac{x+1}{x^{4}+6x^{2}+4x}dx$

This gives a cubic polynomial in denominator and nonfactorisable It looks so simple but i just am not able to solve it
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0answers
67 views

Riemannian generalization/adaption of the Hubbard–Stratonovich transformation

I'd like to write the Hubbard–Stratonovich (HS) transformation of a scalar function on a Riemannian manifold (curved space). This transformation is quite simple in Euclidean space. One can consider ...
2
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0answers
242 views

Contour Integral solution to differential equations, Euler transformation?

In Spain's book, Functions of mathematical physics he introduces the contour integral method of solving ODEs. The baseic idea is: given an ODE $\sum_0^m a_r(t) \frac {d^rf}{dt^r} = 0$, a solution may ...
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1answer
114 views

Extending the domains of densely defined bounded integral transforms on $L^2(\Bbb R)$

This is a question I've contemplated for quite some time since it's pretty closely related to Fourier theory (particularly choosing the "right" space to define the Fourier transform on). However I've ...
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1answer
306 views

What exactly is an integral kernel?

I am not sure if I have seen integral transforms in the right way, but given a transform like Fourier transform - it's actually a basis transformation right ? $$ F(y) = \int K(x,y) f(x) \text{d}x $$ ...
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1answer
51 views

Fourier transform is unitary proof and other unitary integral operators

There is this old unanswered question: Proof the Fourier Transform is Unitary/Not Unitary What is the easiest way to see that the Fourier transform is unitary and why it is important to have constant ...
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1answer
64 views

Finding the root mean square of a sum of trig functions

$$v(t) = 3 - 2\sin(t) + 8\sin^2(t)$$ To find the rms of this function, I first figured out that the period $T = 2\pi$. I then set up the equation: $V = \sqrt{\frac{1}{T}\int^T_0v^2(t)\,dt}$ ...
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0answers
20 views

Conditions for Mellin inversion

Under which conditions is the function $$ g(s)=a^{c(s-1)}\Gamma(s),\qquad a>0,c\in \mathbb{R} $$ the Mellin transform of a probability density function $f$? If $c=-1$, then $f$ is the exponential ...
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0answers
18 views

PDE Variable transform (integral) Chain Rule

Can somebody help me with the variable transform where the new variable is integral variable? This is a moving boundary problem where the radius of particle, $R(t)$ changes with time. The equations ...
2
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1answer
53 views

Hankel transform of a Bessel function of different order

Here I found that $$ \int_0^\infty J_\nu(kr) J_\nu(sr) r dr = \frac{\delta(k - s)}{s} = \frac{1}{s^2}\delta\left(1 - \frac{k}{s}\right). $$ I wonder how can that be derived and if a similar method can ...
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0answers
98 views

Fourier transform of an inverse function.

If for a given function $f(x)$, the Fourier transform is $\hat{f}(p)$; Is there a way to find the Fourier transform of $f(x)^{-1}$ in terms of $\hat{f}(p)$?
3
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1answer
57 views

Norm of an integral operator $L^1 \rightarrow L^\infty$

Let $T:L^1(\mathbb{R}^n)\rightarrow L^\infty(\mathbb{R}^n)$ be an integral operator, i.e. there exist $K:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^n$ such that for all $f\in ...
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1answer
101 views

Antiderivative of $\frac{e^x}{\sqrt{1-x^2}}$

Can anyone help me find the following indefinite integral: $$\int{\frac{e^x}{\sqrt{1-x^2}} dx}$$ I cannot think of any transformation...
3
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0answers
71 views

Integral of an expression involving sine and cosine powers

For integers $a,n\in \mathbb N$, consider the following integral $$ I_n(a) = \frac{(-i)^x}{\pi}\int_0^\pi e^{i\theta(n-2a)} \sin^x \theta \cos^{n-x} \theta\; \mathrm d\theta\;. $$ How would one go ...
9
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141 views

What plays the role of the identity for the generalized convolution associated to the Fourier-Bessel transform?

In traditional Fourier theory, the Dirac delta plays the role of an "identity" for the $L^1$ algebra with respect to the usual convolution. The convolution is traditionally built out of group ...
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1answer
35 views

Reconstruction of a function from its moments

The n-th moment of a real valued function $f$ is defined as: $m_n(f)=\int_{-\infty}^{+\infty}x^nf(x)dx$. I heard that a function $f$ is uniquely determined by its moments. I would be quite surprised ...
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1answer
49 views

Dirichlet energy and Fourier transform

Is there a direct relationship between the Dirichlet energy of a function: $$E(f)=\int_{\Omega}\lvert\nabla f(\mathbf{x})\rvert^2\mathrm{d}V$$ and its Fourier transform ...
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6answers
1k views

What does it mean when two functions are “orthogonal”, why is it important?

I have often come across the concept of orthogonality and orthogonal functions e.g in fourier series the basis functions are cos and sine, and they are orthogonal. For vectors being orthogonal means ...
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1answer
28 views

Two small questions concerning the Hankel transform

For an application, I was reading the wikipedia page on the Hankel transform and I was hoping somebody could clarify two things for me that I have not been able to find elsewhere as well: 1) ...
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0answers
32 views

2D Poisson equation and Bessel Functions

If I were to solve a inhomogeneous 2D Laplace's equation (in polar coordinates), of the form: $$\nabla^2 f= J_2(r)$$ How would I use the Hankel or the Mellin transform to find a solution? I couldn't ...
1
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1answer
42 views

Is the Fourier Transform of the limit the limit of the Fourier Transform?

Assume you want to compute the Fourier transform of a function $f_\epsilon(x)$ given by \begin{align} \mathcal{F}(f_\epsilon)(k) = \int f_\epsilon(x) e^{-ikx}\, dx \end{align} Further assume, that ...
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5answers
61 views

“eigenfunction” of a transformation

Fourier transform of a gaussian is another gaussian. Fourier/Laplace transforms of $\frac{1}{\sqrt t}$ is something like $\frac{1}{\sqrt \omega}$. I realize that we can't call these eigenfunctions ...
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2answers
68 views

trouble with non-homogeneous ODE system… which method shall I use?

I am an undergrad statistics student and I am having troubles with non-homogeneous ODE systems. During my classes I went over just three methods for solving odes: Laplace transform, Fourier transform ...
1
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1answer
50 views

Smoothness of Fourier transform of $\frac{1}{|x|^p}$

Consider the "function" (more precisely it is a tempered distribution) given by $f : \mathbb{R}^n \to \mathbb{R}$, $f(x) = \frac{1}{|x|^p}$, where $0 < p < n$. It can be calculated that the ...
2
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0answers
68 views

Inverse Fourier transform of $\frac{1}{\sqrt{\xi}}$

Does talking about the inverse Fourier transform of $\frac{1}{\sqrt{\xi}}$ even make sense? If it does, how can we conclude about the decay properties, support and smoothness of the inverse Fourier ...
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1answer
42 views

Identifying a function is even or odd or not even and odd. [closed]

Here I have a very confusing problem. I'm right now solving Fourier transform. In which different formulas has to be applied according to the nature of the function wether it is odd or even or not ...
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1answer
28 views

Probability transformation

I have a question regarding probability transformations. Could someone tell me wether I am doing it good or not? Consider $f_{X}(x)=3/2e^{-3x}+3e^{-6x}, x \geq 0$, 1) Calculate the pdf of ...
3
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3answers
218 views

A more elegant way to find the Fourier transform

Let $f$ be defined analytically as : $$f(x)=\arccos \left ( \sin \left ( 2x \right ) \right ), x \in\left (0,10 \right ], f(x)=0, x\notin\left ( 0,10 \right ]$$ Here is a graph of the above ...
3
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1answer
63 views

Inverse Radon Transform of a function in the Schwartz class

This question comes from reading through Stein and Shakarchi's Fourier Analysis, page 206. Consider the two Schwartz spaces $\mathcal{S}(\mathbb{R}^3)$ and $\mathcal{S}(\mathbb{R}\times S^2)$, where ...
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0answers
76 views

Geometric Interpretation of Fourier Transforms

I'm interested in tsunami wave science and I've already got an engineering degree and a basic knowledge of signal processing. The courses I took were intensively computational and taught some skills ...
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0answers
22 views

Consistency/range conditions for (integral) transform mapping into higher-dimensional space

I am interested in learning more about what the (formal) implications are when transforming a $n$-dimensional function space (e.g., the space of all $\mathbb{R}^n\to\mathbb{R}$) to a higherdimensional ...
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29 views

Singularities in the Gauss Hypergeometric Function

I am evaluating the following term in a series: $$I_k = \int\!x^{-3(2k+1)}(1+\lambda x^4)^{-1/2}\,\mathrm dx$$ When I plug this into WolframAlpha, I get the following result: $$I_k = ...